de Rham cohomology

de Rham 上同调

Geometry & Topology · Graduate · Mathematics Resources editorial compilation (AI-assisted)

Translation evidence: Literature usage

Meaning in context

For a smooth manifold, the real vector spaces obtained in each degree by dividing the closed smooth differential forms by the exact ones, using the exterior derivative; degree zero has no incoming differential.

对光滑流形，逐次数将闭的光滑微分形式空间除以恰当形式空间所得的实向量空间；微分为外微分，零次没有流入的微分。

Example

On $S^1$, $\eta=x\,dy-y\,dx$ has integral $2\pi$. Hence it is closed but cannot equal $df$ for a global function, giving a nonzero class in $H^1_{\mathrm{dR}}(S^1)$.

Usage and distinctions

Local primitives do not supply a global primitive. Integral singular cohomology can have torsion, whereas these groups use real coefficients.

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Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Liviu I. Nicolaescu — Lectures on the Geometry of Manifolds§7.1.3, Very little homological algebra, Example 7.1.9, The DeRham complex · Checked 2026-10-14

Translation sources

1. Li Xiaobin — Southwest Jiaotong University, Gromov–Witten study guide§1.1 微分流形，核心内容：de Rham 上同调 · Checked 2026-10-14

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